Factor of similitude group for a bilinear form

Definition

Let $k$ be a field, $V$ a (usually finite-dimensional) vector space over $k$, and $b$ a bilinear form on $V$. The factor of similitude group for $b$ is the subgroup of the multiplicative group of $k$ comprising those $\lambda$ such that there exists an invertible linear transformation $A$ satisfying:

$b(Av,Aw)=\lambda b(v,w)\ \forall \ v,w\in V$.

We typically assume $b$ to be nondegenerate, though this is not necessary to make sense of the definition.

Facts

• All squares are in the factor of similitude group, because if $\alpha$ is in the multiplicative group of $k$, then the factor of similitude for scalar multiplication by $\alpha$ is $\alpha ^{2}$
• For $b$ nondegenerate, the factor of similitude group is contained in the subgroup comprising $n^{th}$ roots of squares, because the determinant (which is the $n^{th}$ power of the factor of similitude) must be a square.